Theorem xrmaxleim 11387

Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)

Assertion

Ref	Expression
xrmaxleim	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Proof of Theorem xrmaxleim

Dummy variables f g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 9863	. . . 4 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		simplr 528	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. RR* )
4		prid2g 3723	. . . 4 |- ( B e. RR* -> B e. { A , B } )
5	3, 4	syl 14	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. { A , B } )
6		simpllr 534	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A <_ B )
7		xrlenlt 8084	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
8	7	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( A <_ B <-> -. B < A ) )
9	6, 8	mpbid 147	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < A )
10		breq2 4033	. . . . . . 7 |- ( y = A -> ( B < y <-> B < A ) )
11	10	notbid 668	. . . . . 6 |- ( y = A -> ( -. B < y <-> -. B < A ) )
12	11	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( -. B < y <-> -. B < A ) )
13	9, 12	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < y )
14		xrltnr 9845	. . . . . 6 |- ( B e. RR* -> -. B < B )
15	14	ad4antlr 495	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < B )
16		breq2 4033	. . . . . . 7 |- ( y = B -> ( B < y <-> B < B ) )
17	16	notbid 668	. . . . . 6 |- ( y = B -> ( -. B < y <-> -. B < B ) )
18	17	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> ( -. B < y <-> -. B < B ) )
19	15, 18	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < y )
20		elpri 3641	. . . . 5 |- ( y e. { A , B } -> ( y = A \/ y = B ) )
21	20	adantl 277	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> ( y = A \/ y = B ) )
22	13, 19, 21	mpjaodan 799	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> -. B < y )
23	2, 3, 5, 22	supmaxti 7063	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> sup ( { A , B } , RR* , < ) = B )
24	23	ex 115	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   {cpr 3619   class class class wbr 4029   supcsup 7041   RR*cxr 8053   < clt 8054   <_ cle 8055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltirr 7984  ax-pre-apti 7987

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-iota 5215  df-riota 5873  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060

This theorem is referenced by:  xrmaxltsup  11401  xrmaxadd  11404  xrmineqinf  11412